Method of inverting nearly Toeplitz or block Toeplitz matrices

ABSTRACT

A method of computing an inversion (X) of a nearly Toeplitz n by n matrix (A). A perturbation matrix (E) is first determined such that the sum of the nearly Toeplitz matrix (A) and the perturbation matrix (E) is a Toeplitz matrix (T). The inversion is solved by solving the equation X=T −1 (B+EX), where B is a vector or matrix of dimension n by m. An initial estimate X (0)  is selected and estimates of the inversion X are iteratively computed through the recursion X (n−1) =T −1 (B+EX (n) ). The initial estimate X (0)  may be equal to an inversion (T −1 ) of the Toeplitz matrix (T). The present invention may be utilized in a radio receiver to efficiently compute (1) a least-squares (LS) channel estimate, (2) minimum mean squared error (MMSE) prefilter coefficients for a decision feedback equalizer (DFE), or (3) an autoregressive (AR) noise-spectrum estimation from a finite number of observed noise samples.

BACKGROUND OF THE INVENTION

The present invention relates generally to radio telecommunicationnetworks. More particularly, and not by way of limitation, the presentinvention is directed to an efficient method of inverting nearlyToeplitz or block Toeplitz matrices in adaptive wireless receivers.

The ability to adapt to different propagation and interferenceconditions is highly important in modern wireless receivers. In atypical adaptive receiver, various parameters that characterize thecurrent communication link conditions, such as channel response andinterference statistics, are often estimated over a limited number ofreceived signal samples. The core of such estimation processes oftenrequires the inversion of matrices that have approximately Toeplitz orblock Toeplitz forms. A Toeplitz matrix is any n×n matrix with valuesconstant along each top-left-to-lower-right diagonal. More precisely, ifT is a Toeplitz matrix, then the element in the i-th row and the j-thcolumn, for any i and j, can be expressed as [T_(ij)]=a(i-j) for somescalar function, a(k), where k is an integer, as shown below:

$T = \begin{bmatrix}{a(0)} & {a\left( {- 1} \right)} & \ldots & {a\left( {- \left( {n - 2} \right)} \right)} & {a\left( {- \left( {n - 1} \right)} \right)} \\{a(1)} & {a(0)} & {a\left( {- 1} \right)} & \vdots & {a\left( {- \left( {n - 2} \right)} \right)} \\\vdots & {a(1)} & {a(0)} & {a\left( {- 1} \right)} & \vdots \\{a\left( {n - 2} \right)} & \vdots & {a(1)} & {a(0)} & {a\left( {- 1} \right)} \\{a\left( {n - 1} \right)} & {a\left( {n - 2} \right)} & \ldots & {a(1)} & {a(0)}\end{bmatrix}$A block Toeplitz matrix is defined in the same manner except that a(k)is a p×p matrix function where n is divisible by the integer p.

An exemplary use of nearly Toeplitz or block Toeplitz matrices is foundin equalizers that address the problem of multipath fading of wirelesschannels. Multipath fading is a key phenomenon that makes reliabletransmission difficult in a wireless communication system, and istypically caused by reflections of the transmitted radio signal fromnumerous local scattering objects situated around the receiver. Not onlyis such multi-path fading time-varying, but it also causes intersymbolinterference (ISI) at the receiver. To mitigate the detrimental effectof ISI, a wireless receiver typically uses an equalizer. One of the bestand most popular equalization method is the maximum-likelihood sequenceestimation (MLSE). To operate properly, an MLSE equalizer requires anaccurate estimate of the wireless channel. Because of the time-varyingnature of the fading channel, the channel estimate often needs to beupdated periodically. The least-squares (LS) method is one of the mostcommonly used methods of channel estimation. To compute an LS channelestimate, it is known that the inversion of so-called Fisherinformation, which is a matrix that is nearly, but not exactly Toeplitz,is required. (See, Crozier, S. N. et. al., “Least sum of squared errors(LSSE) channel estimation,” IEEE Proceedings-F 1991, pp. 371-378, whichis hereby incorporated by reference.)

Although an MLSE equalizer can provide excellent performance in terms ofreducing the overall bit error rates, its complexity of implementationgrows exponentially with the length of the channel estimate.Consequently, in situations where the channel response is relativelylong, a reduced complexity equalizer, such as the decision-feedbackequalizer (DFE) is often used instead. A DFE equalizer demodulates thetransmitted symbols sequentially one after another and uses thedemodulated symbols of previous time instants to estimate the effect ofISI that these previous symbols cause at the next time instant. In orderto improve the estimate of ISI, a prefilter is often used before a DFEequalizer to transform the effective channel response so that most ofthe energy of the resulting channel response, after prefiltering, isconcentrated in the front-most channel tap. Since the original channelresponse may vary with time, the prefilter also needs to be computedperiodically. One of the most commonly used prefilters is theminimum-mean-squared-error (MMSE) prefilter. To compute an MMSEprefilter, the receiver needs to invert a nearly Toeplitz matrix thatdepends on the channel response as well as the variance of noise in thereceived signal. (See, Proakis, John G., Digital Communications, 2^(nd)edition, McGraw-Hill, 1989, which is hereby incorporated by reference.)

For yet another example, consider a high-capacity cellular communicationnetwork where radio frequencies are being reused in differentgeographical areas within close proximity. In such a network, mutualinterference among users occupying the same radio channel is often amajor source of signal disturbance. Thus, mobile receivers that arecapable of suppressing interference are highly desirable. A simple andeffective method of suppressing interference is to model theinterference as colored noise and to attempt to whiten the noise using alinear predictive filter, commonly referred to as a whitening filter.The whitening filter flattens the frequency spectrum of the noise (or“whitens” the noise) by subtracting from it the portion that ispredictable using estimates of noise samples from previous timeinstants. Through this whitening process, the noise power is reduced.Since the spectrum of the interference typically varies with time, thereceiver often needs to adaptively compute such a whitening filter basedon the received signal. A popular and effective method of computing thiswhitening filter, or equivalently of estimating the interferencespectrum, is the covariance method. (See, Kay, S. M., Modern SpectralEstimation, Theroy & Application, Prentice-Hall, 1988, which is herebyincorporated by reference.) The covariance method requires thecalculation of the inverse of a noise covariance matrix that is nearly,but not exactly Toeplitz. This covariance matrix becomes a blockToeplitz matrix when the received signal consists of multiple branches,which may come from multiple antennas or oversampling of the receivedsignal.

As can be seen from the above examples, in channel estimation processes,these matrices are typically not known a priori, and since they need tobe inverted in real time, efficient techniques for inverting nearlyToeplitz matrices are highly desirable. While efficient algorithms existfor inverting Toeplitz and block Toeplitz matrices, relatively little isknown about inverting nearly Toeplitz matrices. Typically, nearlyToeplitz matrices have been inverted using techniques for solving linearequations such as the Gaussian elimination, Cholesky decomposition, andthe Gauss-Seidel algorithm. However, these techniques do not exploit thenearly Toeplitz structure, and thus tend to be too complex to implementefficiently. In Friedlander, B., et al., “New Inversion Formulas forMatrices Classified in Terms of Their Distance From Toeplitz Matrices,”Linear Algebra and its Applications, Vol. 27, pp. 31-60, 1979, analgorithm was proposed for inverting a class of nearly Toeplitz matricesbased on the notion of displacement rank. This algorithm is quitecomplex to implement and provides computational benefits only for thosematrices with low displacement ranks. In co-owned PCT application WO02/15505, an algorithm is proposed for inverting a specific form ofnearly Toeplitz matrix of the form, (C^(H)C+σ²I), where C is anL_(f)×L_(f) truncated convolutional matrix of the channel response.However, the algorithm proposed in WO 02/15505 is also complex and onlyworks for matrices in this specific form, but not for other nearlyToeplitz matrices in general.

BRIEF SUMMARY OF THE INVENTION

In one aspect, the present invention is directed to a simple andeffective method of computing an inversion (X) of a nearly Toeplitz n byn matrix (A), the inversion (X) being a matrix of dimensions n by m. Themethod includes determining a perturbation matrix (E) such that the sumof the nearly Toeplitz matrix (A) and the perturbation matrix (E) is aToeplitz matrix (T). The inversion is then solved for by solving theequation X=T⁻¹(B+EX), where B is a vector or matrix of dimension n by m.The solving step includes the steps of selecting an initial estimateX⁽⁰⁾; and iteratively computing estimates of the inversion X through therecursion X^((n+1))=T⁻¹(B+EX^((n))).

In specific embodiments, the present invention may be utilized in aradio receiver to efficiently compute (1) a least-squares (LS) channelestimate, (2) minimum mean squared error (MMSE) prefilter coefficientsfor a decision feedback equalizer (DFE), or (3) an autoregressive (AR)noise-spectrum estimation from a finite number of observed noisesamples.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the following section, the invention will be described with referenceto exemplary embodiments illustrated in the figures, in which:

FIG. 1 is a flow chart illustrating the steps of an exemplary embodimentof the method of the present invention;

FIG. 2 is a simplified block diagram of a portion of an adaptivereceiver illustrating where a least-squares channel estimate isperformed in accordance with the teachings of the present invention;

FIG. 3 is a simplified block diagram of a portion of an adaptivereceiver illustrating where a channel estimation is used to calculateMMSE prefilter coefficients for use in a Decision Feedback Equalizer(DFE) in accordance with the teachings of the present invention;

FIG. 4 is a simplified block diagram of a portion of an adaptivereceiver illustrating where an autoregressive (AR) noise-spectrumestimation is determined in accordance with the teachings of the presentinvention;

FIG. 5 is a graph of relative difference as a function of number ofiterations of the algorithm of the present invention and theGauss-Seidel algorithm; and

FIG. 6 is a graph of block error rate as a function ofcarrier-to-interference (C/I) ratio for a two-antenna receiver utilizingdiffering numbers of iterations of the algorithm of the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

This disclosure describes an iterative algorithm for inverting nearlyToeplitz matrices that can be utilized, for example, in many areas ofbaseband receiver algorithms. In summary, the invention addresses theproblem of solvingAX=B  (1)where A is an n by n invertible, nearly Toeplitz (or block Toeplitz)matrix, B is a vector or matrix of dimension n by m, and X is thedesired inverse or solution of dimensions n by m.

FIG. 1 is a flow chart illustrating the steps of an exemplary embodimentof the method of the present invention. The method starts at step 11 byidentifying a nearly Toeplitz (or block Toeplitz) n by n matrix A. Atstep 12, a “perturbation” matrix E is determined such that A+E=T, whereT is a Toeplitz (or block Toeplitz) matrix. At step 13, the inversematrix of T is computed using known efficient algorithms for invertingToeplitz or block Toeplitz matrix. If A is nearly Toeplitz or blockToeplitz, then E is “small” in some sense (for example, its L₂ norm issmall). It follows that equation (1) can be written asX=T ⁻¹(B+EX)  (2)

At step 14, the fixed point equation (2) is solved by iterativelycomputing estimates of the actual inverse X through the recursionX ^((n+1)) =T ⁻¹(B+EX ^((n)))  (3)starting with an arbitrary initial estimate X⁽⁰⁾. For example, X⁽⁰⁾ maybe set equal to the inversion of the Toeplitz matrix (T⁻¹). If T is aToeplitz matrix, T⁻¹ can be efficiently computed using known algorithms,such as the Trench or the Strang algorithm. The Trench algorithm isdescribed in Trench, W. F., “An Algorithm for the Inversion of FiniteToeplitz Matrices”, J. SIAM 12, pp. 515-522, September 1964. The Strangalgorithm is described in Strang, G., “A Proposal for Toeplitz MatrixCalculations”, Stud. Apply Math., pp. 171-176, January 1976. If T isblock Toeplitz, the Akaike algorithm, which is a generalization of theTrench algorithm to the block Toeplitz case, can be used. The Akaikealgorithm is described in Akaike, H., “Block Toeplitz matrix inversion,”SIAM J. Appl. Math., vol. 24, pp. 234-241, March 1973. All of thesereferences are incorporated herein by reference. The recursion inequation (3) is guaranteed to converge so long as the maximum eigenvalueof T⁻¹E is less than one, which is the case when A is nearly Toeplitz.

At step 15, it is determined whether the desired level of accuracy forthe inversion has been achieved. If not, the method returns to step 14and performs another iteration. If the desired level of accuracy hasbeen achieved, the method moves to step 16 where the result is sent tothe appropriate application for further processing.

When utilized in a baseband receiver algorithm, the perturbation matrixcan often be written in the form of E=ε^(H)ε, for a certain matrix ε,which stems from truncated portions of the corresponding convolutionmatrix of each problem. As an illustrative example, consider the widelyused least-squares (LS) channel estimate ĉ over a sequence of knownsymbols

{s[n]}_(k − 0)^(N − 1)given byĉ=(S ^(H) S)⁻¹ S ^(H) r  (4)where r is a vector of received samples and S is a truncated convolutionmatrix of the transmitted symbols given by

$\begin{matrix}{S \equiv {\begin{bmatrix}{s\left\lbrack {L - 1} \right\rbrack} & {s\left\lbrack {L - 2} \right\rbrack} & \ldots & {s\lbrack 0\rbrack} \\{s\lbrack L\rbrack} & {s\left\lbrack {L - 1} \right\rbrack} & \ldots & {s\lbrack 1\rbrack} \\\vdots & \vdots & \ldots & \vdots \\{s\left\lbrack {N - 1} \right\rbrack} & {s\left\lbrack {N - 2} \right\rbrack} & \ldots & {s\left\lbrack {N - L} \right\rbrack}\end{bmatrix}.}} & (5)\end{matrix}$

Because of the truncated nature of S, the Fisher information matrixS^(H)S that needs to be inverted in equation (4) is approximately, butnot exactly, Toeplitz. When the transmitted symbols are known a priori,the inverse of S^(H)S can be pre-computed to minimize the efforts neededto compute ĉ in real time. However, in high-performance receivers, thereceived samples r may be filtered by a certain noise-whitening filterbefore channel estimation. In this case, the Fisher information matrixS^(H)S is based on whitened symbols (i.e., transmitted symbols filteredby the same noise whitening filter), which are unknown a priori andtherefore must be computed in real time.

For the LS channel estimation problem, the matrix ε is given by:

$\begin{matrix}{ɛ \equiv {\begin{bmatrix}{s\lbrack 0\rbrack} & 0 & \ldots & 0 \\{s\lbrack 1\rbrack} & {s\lbrack 0\rbrack} & \ldots & 0 \\\vdots & \vdots & \ldots & \vdots \\{s\left\lbrack {L - 2} \right\rbrack} & \ldots & {s\lbrack 0\rbrack} & 0 \\0 & {s\left\lbrack {N - 1} \right\rbrack} & \ldots & {s\left\lbrack {N - L + 1} \right\rbrack} \\\vdots & \vdots & \ldots & \vdots \\0 & \ldots & {s\left\lbrack {N - 1} \right\rbrack} & {s\left\lbrack {N - 2} \right\rbrack} \\0 & \ldots & 0 & {s\left\lbrack {N - 1} \right\rbrack}\end{bmatrix}.}} & (6)\end{matrix}$

FIG. 2 is a simplified block diagram of a portion of an adaptivereceiver illustrating where a least-squares channel estimate isperformed in accordance with the teachings of the present invention.Received and downsampled samples, r[n], 21 are input to a channelestimation function 22 where the process described above is utilized todetermine the least-squares (LS) channel estimate (ĉ) 23. The receivedsamples, r[n], and the channel estimate, ĉ, are then provided to anequalization function 24.

As another example, consider the computation of minimum mean squarederror (MMSE) prefilter coefficients for a decision feedback equalizer(DFE). The vector of the MMSE prefilter coefficients of length L_(f),denoted by f, is given byf=(C ^(H) C+σ ² I)⁻¹ C ^(H) e _(L) _(f)   (7)where C is an L_(f)×L_(f) truncated convolutional matrix of the channelresponse given by

$\begin{matrix}{C \equiv {\begin{bmatrix}{c\lbrack 0\rbrack} & 0 & \ldots & 0 & 0 & \ldots & 0 \\{c\lbrack 1\rbrack} & {c\lbrack 0\rbrack} & \ldots & 0 & 0 & \ldots & 0 \\{c\lbrack 2\rbrack} & {c\lbrack 1\rbrack} & {c\lbrack 0\rbrack} & 0 & 0 & \ldots & 0 \\\vdots & \vdots & \ldots & \vdots & \; & \; & \; \\0 & \ldots & 0 & {c\left\lbrack {L - 1} \right\rbrack} & \ldots & {c\lbrack 1\rbrack} & {c\lbrack 0\rbrack}\end{bmatrix}.}} & (8)\end{matrix}$

{c[k]}_(k = 0)^(L − 1)denotes an L-tap channel response, σ denotes the variance of noise inthe received signal, I is an L_(f)×L_(f) identity matrix, and e_(Lf) isthe last column of an L_(f)×L_(f) identity matrix. Note that the matrix(C^(H)C+σ²I) is again, approximately Toeplitz.

For the MMSE prefilter problem, the matrix ε is given by:

$\begin{matrix}{ɛ \equiv {\begin{bmatrix}0 & {c\left\lbrack {L - 1} \right\rbrack} & \ldots & {c\lbrack 1\rbrack} \\\vdots & \vdots & \ldots & \vdots \\0 & \ldots & {c\left\lbrack {L - 1} \right\rbrack} & {c\left\lbrack {L - 2} \right\rbrack} \\0 & \ldots & 0 & {c\left\lbrack {N - 1} \right\rbrack}\end{bmatrix}.}} & (9)\end{matrix}$

FIG. 3 is a simplified block diagram of a portion of an adaptivereceiver illustrating where a channel estimation is used to calculateMMSE prefilter coefficients for use in a Decision Feedback Equalizer(DFE) in accordance with the teachings of the present invention.Received and downsampled samples, r[n], 21 are input to a channelestimation function 22 where the process described above is utilized todetermine the least-squares (LS) channel estimate (ĉ) 23 and a noisevariance estimate ({circumflex over (σ)}) 25. The channel estimate, ĉ,is then provided to a prefilter computation function 31 where theprocess described immediately above is utilized to determine prefiltercoefficients (f) 32. The received samples, r[n], and the prefiltercoefficients, f, are then provided to a prefiltering function 33. Theresults of the prefiltering are then provided to a DFE equalizationfunction 34.

As a third example, consider the problem of autoregressive (AR)noise-spectrum estimation from a finite number of observed noise samples

{y[n]}_(n = 0)^(N − 1).In typical adaptive receivers, these noise samples may represent theresidual samples obtained by subtracting the hypothesized receivedsamples constructed based on a channel estimate from the actual receivedsamples. A common method of estimating the AR coefficients that mostaccurately fit the spectrum of

{y[n]}_(n = 0)^(N − 1)is the covariance method. In this method, the vector of AR coefficients,denoted by a=(a₁, a₂, . . . , a_(M)) is computed by:

$\begin{matrix}{{a = {\left( {\sum\limits_{n = M}^{N - 1}{{y\lbrack n\rbrack}{y\left\lbrack {n - 1} \right\rbrack}^{H}}} \right){\underset{\underset{M^{- 1}}{︸}}{\left( {\sum\limits_{n = M}^{N - 1}{{y\left\lbrack {n - 1} \right\rbrack}{y\left\lbrack {n - 1} \right\rbrack}^{H}}} \right)}}^{- 1}}},} & (10)\end{matrix}$where y[n]≡(y[n], y[n−1], . . . , y[n−M+1])^(T). Similar to the previoustwo examples, the matrix M that needs to be inverted in this problem isagain approximately, but not exactly, Toeplitz. (In the case formultiple-antenna receivers the matrix M is an almost block Toeplitzmatrix.)

For the AR noise-spectrum estimation problem, the matrix ε is given by:

$\begin{matrix}{ɛ \equiv {\begin{bmatrix}{y\lbrack 0\rbrack} & 0 & \ldots & 0 \\{y\lbrack 1\rbrack} & {y\lbrack 0\rbrack} & \ldots & 0 \\\vdots & \vdots & \ldots & \vdots \\{y\left\lbrack {M - 2} \right\rbrack} & \ldots & {y\lbrack 0\rbrack} & 0 \\0 & {y\left\lbrack {N - 1} \right\rbrack} & \ldots & {y\left\lbrack {N - M + 1} \right\rbrack} \\\vdots & \vdots & \ldots & \vdots \\0 & \ldots & {y\left\lbrack {N - 1} \right\rbrack} & {y\left\lbrack {N - 2} \right\rbrack} \\0 & \ldots & 0 & {y\left\lbrack {N - 1} \right\rbrack}\end{bmatrix}.}} & (11)\end{matrix}$

FIG. 4 is a simplified block diagram of a portion of an adaptivereceiver illustrating where an autoregressive (AR) noise-spectrumestimation is determined in accordance with the teachings of the presentinvention. Received and downsampled samples, r[n], 21 are input to achannel estimation function 22 where the process described above isutilized to determine the least-squares (LS) channel estimate (ĉ) 23.The channel estimate, ĉ, is then utilized in a noise generation function41 to generate noise samples y[n] 42. The noise samples, y[n], are thenutilized in a whitening filter computation function 43 to determine ARfilter coefficients (a) 44. The received samples, r[n], and the ARfilter coefficients, a, are then provided to a whitening filter 45. Theresults of the filtering are then provided to an equalization function46.

The invention may be implemented, for example, in receivers utilized inthe Global System for Mobile Communications (GSM) and Enhanced Data forGSM Evolution (EDGE) systems. For example, consider the LS channelestimate problem described above. The invention provides an efficientmethod to compute more reliable channel estimates using whitenedreceived samples with whitened training sequences. FIG. 5 shows, aftereach iteration of the algorithm of the present invention, the accuracyof inverses of the Fisher matrices of whitened training sequences. Theaccuracy is measured by the percentage difference in Euclidean-norm,i.e.

$\begin{matrix}{{Error} = {\frac{{X^{(n)} - X}}{X}.}} & (12)\end{matrix}$

Both cases of L=4 and L=7 are shown. For comparison, the accuracy of theinverse computed by the conventional Gauss-Seidel algorithm is alsoshown. As shown in FIG. 5, the algorithm of the present inventionconverges much faster than the Gauss-Seidel algorithm because theinvention exploits the nearly Toeplitz structure of the Fisher matrices.Only two iterations in the 4-tap case and four iterations in the 7-tapcase are needed to bring the error below 30 dB.

FIG. 6 shows the block error rate performance when the present inventionis used to estimate the channel response after whitening filtering in a2-antenna receiver. The 0-th iteration means that T⁻¹ is directly usedas the inverse of the Fisher information matrix in equation (4). Asshown in FIG. 6, only one or two iterations suffice to obtain the sameperformance as that provided by an exact inversion, because theinvention exploits the nearly block Toeplitz structure of the underlyingFisher matrix.

As will be recognized by those skilled in the art, the innovativeconcepts described in the present application can be modified and variedover a wide range of applications. Accordingly, the scope of patentedsubject matter should not be limited to any of the specific exemplaryteachings discussed above, but is instead defined by the followingclaims.

1. A method of computing, in a baseband radio receiver, a least-squares (LS) channel estimate, ĉ, over a sequence of known transmitted symbols {s[n]}_(k − 0)^(N − 1) given by the equation: ĉ=(S ^(H) S)⁻¹ S ^(H) r where S^(H)S is a Fisher information matrix, r is a vector of received samples, and S is a truncated convolution matrix of the transmitted symbols given by: ${S \equiv \begin{bmatrix} {s\left\lbrack {L - 1} \right\rbrack} & {s\left\lbrack {L - 2} \right\rbrack} & \ldots & {s\lbrack 0\rbrack} \\ {s\lbrack L\rbrack} & {s\left\lbrack {L - 1} \right\rbrack} & \ldots & {s\lbrack 1\rbrack} \\ \vdots & \vdots & \ldots & \vdots \\ {s\left\lbrack {N - 1} \right\rbrack} & {s\left\lbrack {N - 2} \right\rbrack} & \ldots & {s\left\lbrack {N - L} \right\rbrack} \end{bmatrix}},$ wherein the information matrix S^(H)S is a nearly Toeplitz matrix based on whitened symbols that are unknown a priori and therefore must be computed in real time, said method comprising the steps of: determining a perturbation matrix (E) in the form of E=ε^(H)ε, for a certain matrix, ε, which stems from truncated portions of the convolution matrix (S) such that the sum of the nearly Toeplitz information matrix (S^(H)S) and the perturbation matrix (E) is a Toeplitz matrix (T); solving for the inversion (X) of the nearly Toeplitz matrix (S^(H)S) by solving the equation: X=T ⁻¹(B+EX), where B is a vector or matrix of dimension n by m, said solving step including the steps of: selecting an initial estimate X⁽⁰⁾; iteratively computing estimates of the inversion X through the recursion: X ^((n+1)) =T ⁻¹(B+EX ^((n))); and computing said least-squares channel estimate, ĉ, to equalize the received signal using X.
 2. The method of claim 1, wherein the matrix ε is given by: $ɛ \equiv {\begin{bmatrix} {s\lbrack 0\rbrack} & 0 & \ldots & 0 \\ {s\lbrack 1\rbrack} & {s\lbrack 0\rbrack} & \ldots & 0 \\ \vdots & \vdots & \ldots & \vdots \\ {s\left\lbrack {L - 2} \right\rbrack} & \ldots & {s\lbrack 0\rbrack} & 0 \\ 0 & {s\left\lbrack {N - 1} \right\rbrack} & \ldots & {s\left\lbrack {N - L + 1} \right\rbrack} \\ \vdots & \vdots & \ldots & \vdots \\ 0 & \ldots & {s\left\lbrack {N - 1} \right\rbrack} & {s\left\lbrack {N - 2} \right\rbrack} \\ 0 & \ldots & 0 & {s\left\lbrack {N - 1} \right\rbrack} \end{bmatrix}.}$
 3. The method of claim 1, further comprising, after each iteration of the solving step, the steps of: determining whether a desired level of accuracy has been achieved for the inversion (X) of the nearly Toeplitz matrix (S^(H)S); and when the desired level of accuracy has not been achieved, performing another iteration of the solving step.
 4. The method of claim 1, wherein the step of determining a perturbation matrix (E) includes determining the perturbation matrix (E) such that the maximum eigenvalue of T⁻¹E is less than one, thereby guaranteeing that the recursion in the solving step converges.
 5. A method of computing, in a baseband radio receiver, feedback equalizer (DFE), wherein a vector of the MMSE prefilter coefficients of length L_(f), denoted by f, is given by: f=(C ^(H) C+σ ² I)⁻¹ C ^(H) e _(L) _(f) , where C is an L_(f)×L_(f) truncated convolutional matrix of the channel response given by: ${C \equiv \begin{bmatrix} {c\lbrack 0\rbrack} & 0 & \ldots & 0 & 0 & \ldots & 0 \\ {c\lbrack 1\rbrack} & {c\lbrack 0\rbrack} & \ldots & 0 & 0 & \ldots & 0 \\ {c\lbrack 2\rbrack} & {c\lbrack 1\rbrack} & {c\lbrack 0\rbrack} & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \ldots & \vdots & \; & \; & \; \\ 0 & \ldots & 0 & {c\left\lbrack {L - 1} \right\rbrack} & \ldots & {c\lbrack 1\rbrack} & {c\lbrack 0\rbrack} \end{bmatrix}},$ and where (C^(H)C+σ²I) is a nearly Toeplitz n by n matrix, and e_(L) _(f) is the last column of an L_(f)×L_(f) identity matrix, said method comprising the steps of: determining a perturbation matrix (E) in the form of E=ε^(H)ε, for a certain matrix, ε, which stems from truncated portions of the convolution matrix (C) such that the sum of the nearly Toeplitz matrix (C^(H)C+σ²I) and the perturbation matrix (E) is a Toeplitz matrix (T); solving for the inversion (X) of the nearly Toeplitz matrix (C^(H)C+σ²I) by solving the equation: X=T ⁻¹(B+EX), where B is a vector or matrix of dimension n by m, said solving step including the steps of: selecting an initial estimate X⁽⁰⁾; iteratively computing estimates of the inversion X through the recursion: X ^((n+1)) =T ⁻¹(B+EX ^((n))); and computing said minimum mean squared error prefilter coefficients to equalize the received signal using X.
 6. The method of claim 5, wherein the matrix ε is given by: $ɛ \equiv {\begin{bmatrix} 0 & {c\left\lbrack {L - 1} \right\rbrack} & \Lambda & {c\lbrack 1\rbrack} \\ M & M & \Lambda & M \\ 0 & \Lambda & {c\left\lbrack {L - 1} \right\rbrack} & {c\left\lbrack {L - 2} \right\rbrack} \\ 0 & \Lambda & 0 & {c\left\lbrack {N - 1} \right\rbrack} \end{bmatrix}.}$
 7. The method of claim 5, further comprising, after each iteration of the solving step, the steps of: determining whether a desired level of accuracy has been achieved for the inversion (X) of the nearly Toeplitz matrix (C^(H)C+σ²I); and when the desired level of accuracy has not been achieved, performing another iteration of the solving step.
 8. The method of claim 5, wherein the step of determining a perturbation matrix (E) includes determining the perturbation matrix (E) such that the maximum eigenvalue of T⁻¹E is less than one, thereby guaranteeing that the recursion in the solving step converges.
 9. In an adaptive radio receiver, a method of computing an autoregressive (AR) noise-spectrum estimation from a finite number of observed noise samples {y[n]}_(n = 0)^(N − 1), said method comprising the steps of: computing a channel estimate; generating hypothesized received samples based on the channel estimate; determining residual noise samples by subtracting the hypothesized received samples from the actual received samples; estimating AR coefficients that most accurately fit the spectrum of {y[n]}_(n = 0)^(N − 1), wherein a vector of the AR coefficients, denoted by a=(a₁, a₂, . . . , a_(M)), is computed by: ${a = {\left( {\sum\limits_{n = M}^{N - 1}{{y\lbrack n\rbrack}{y\left\lbrack {n - 1} \right\rbrack}^{H}}} \right)\underset{\underset{M^{- 1}}{︸}}{\left( {\sum\limits_{n = M}^{N - 1}{{y\left\lbrack {n - 1} \right\rbrack}{y\left\lbrack {n - 1} \right\rbrack}^{H}}} \right)^{- 1}}}},$ where y[n]≡(y[n], y[n−1], . . . , y[n−M+1])^(T), and M is a nearly Toeplitz n by n matrix, wherein the nearly Toeplitz matrix (M) is inverted by the steps of: determining a perturbation matrix (E) in the form of E=ε^(H)ε, for a certain matrix, ε, such that the sum of the nearly Toeplitz matrix (M) and the perturbation matrix (E) is a Toeplitz matrix (T); solving for the inversion (X) of the nearly Toeplitz matrix (M) by solving the equation: X=T ⁻¹(B+EX), where B is a vector or matrix of dimension n by m, said solving step including the steps of: selecting an initial estimate X⁽⁰⁾; and iteratively computing estimates of the inversion X through the recursion: X ^((n+1)) =T ⁻¹(B+EX ^((n))).
 10. The method of claim 9, wherein the receiver is a multiple-antenna receiver, and the matrix (M) is an almost block Toeplitz matrix.
 11. The method of claim 9, wherein the matrix ε is given by: $ɛ \equiv {\begin{bmatrix} {y\lbrack 0\rbrack} & 0 & \ldots & 0 \\ {y\lbrack 1\rbrack} & {y\lbrack 0\rbrack} & \ldots & 0 \\ \vdots & \vdots & \ldots & \vdots \\ {y\left\lbrack {M - 2} \right\rbrack} & \ldots & {y\lbrack 0\rbrack} & 0 \\ 0 & {y\left\lbrack {N - 1} \right\rbrack} & \ldots & {y\left\lbrack {N - M + 1} \right\rbrack} \\ \vdots & \vdots & \ldots & \vdots \\ 0 & \ldots & {y\left\lbrack {N - 1} \right\rbrack} & {y\left\lbrack {N - 2} \right\rbrack} \\ 0 & \ldots & 0 & {y\left\lbrack {N - 1} \right\rbrack} \end{bmatrix}.}$
 12. The method of claim 9, further comprising, after each iteration of the solving step, the steps of: determining whether a desired level of accuracy has been achieved for the inversion (X) of the nearly Toeplitz matrix (M); and when the desired level of accuracy has not been achieved, performing another iteration of the solving step.
 13. The method of claim 9, wherein the step of determining a perturbation matrix (E) includes determining the perturbation matrix (E) such that the maximum eigenvalue of T⁻¹E is less than one, thereby guaranteeing that the recursion in the solving step converges.
 14. A channel estimator in a radio receiver that computes a least-squares (LS) channel estimate, ĉ, over a sequence of known transmitted symbols {s[n]}_(k − 0)^(N − 1) given by the equation: ĉ=(S ^(H) S)⁻¹ S ^(H) r where S^(H)S is a Fisher information matrix, r is a vector of received samples, and S is a truncated convolution matrix of the transmitted symbols given by: ${S \equiv \begin{bmatrix} {s\left\lbrack {L - 1} \right\rbrack} & {s\left\lbrack {L - 2} \right\rbrack} & \ldots & {s\lbrack 0\rbrack} \\ {s\lbrack L\rbrack} & {s\left\lbrack {L - 1} \right\rbrack} & \ldots & {s\lbrack 1\rbrack} \\ \vdots & \vdots & \ldots & \vdots \\ {s\left\lbrack {N - 1} \right\rbrack} & {s\left\lbrack {N - 2} \right\rbrack} & \ldots & {s\left\lbrack {N - L} \right\rbrack} \end{bmatrix}},$ wherein the information matrix S^(H)S is a nearly Toeplitz matrix based on whitened symbols that are unknown a priori and therefore must be computed in real time, said channel estimator comprising: means for determining a perturbation matrix (E) in the form of E=ε^(H)ε, for a certain matrix, ε, which stems from truncated portions of the convolution matrix (S) such that the sum of the nearly Toeplitz information matrix (S^(H)S) and the perturbation matrix (E) is a Toeplitz matrix (T); means for selecting an initial estimate X⁽⁰⁾ of an inversion X of the nearly Toeplitz information matrix (S^(H)S); means for iteratively computing estimates of the inversion X through the recursion X^((n−1))=T⁻¹(B+EX ^((n))), where B is a vector or matrix of dimension n by m; and means for computing said least-squares channel estimate, ĉ, to equalize the received signal using X.
 15. The channel estimator of claim 14, wherein the means for iteratively computing estimates of the inversion X includes: means for determining whether a desired level of accuracy has been achieved for the inversion (X) of the nearly Toeplitz information matrix (S^(H)S); and means for iteratively computing additional estimates until the desired level of accuracy has been achieved.
 16. The channel estimator of claim 14, wherein the means for determining a perturbation matrix (E) includes means for determining the perturbation matrix (E) such that the maximum eigenvalue of T⁻¹E is less than one, thereby guaranteeing that the recursion in the solving step converges. 